And lies:
(i) On the same side of A as B does
(ii) On the opposite side of A as B does
Now I already know the section formula which is applied when the point is outside the line. I plug in the values and somehow I get different answers. The section formula is given as follows:
(A) When the point lies internally
$$x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2} ,y= \frac{m_1y_2 + m_2y_1}{m_1 + m_2} $$ $Where\ m_1\ and\ m_2\ are \ the\ ratios\ which\ divide\ the\ line.$
(B) When the point lies externally
$$x = \frac{m_1x_2 - m_2x_1}{m_1 - m_2} ,y= \frac{m_1y_2 - m_2y_1}{m_1 - m_2} $$ $Where\ m_1\ and\ m_2\ are \ the\ ratios\ which\ divide\ the\ line.$
As far as I understand the ratio or m and n in the first case are going to 1:2 and in the second part they will be 2:1. But in the solution, the author divides it in 2:1 in the first case and 2:3 in the second case.
And I don't understand why and how? In the first case let's say B is 1 far from A, then P has to be 2 because it has to be twice as far from A as B is. Isn't it safe to say that the ratio then will be 1:2. And in the second case, P has the same condition is only changing sides which I think should simply flip the ratio. Hence being 2:1.
Am I wrong on the way I am assigning ratios or something?
Would appreciate any help on this.
When $P$ is on the same side of $A$ as is $B$,
then $B$ is between $A$ and $P$, the total length of the segment is $AP = AB+BP = 2AB$, and we have $AB:AP = 1:2$. On the other hand, when $P$ is on the opposite side,
then it’s $A$ that’s between the other two points. The total length is $BP=AB+AP=3AB$ and $AP:BP = 2:3$.